16 h In e na ional Symposium on P ac ical Design o Ships and O he Floa ing S uc u es PRADS 2025
Ann A bo , MI, USA, Oc obe 19 h-23 d 2025
Blending Manoeu ing and Seakeeping:
Heading-keeping Simula ions o 5415M
Ruddy Ku nia*, Robe o Tonelli, Nicolas Ca e e, and Tim Bunnik
Ma i ime Resea ch Ins i u e Ne he lands (MARIN), Wageningen, The Ne he lands
Abs ac . This pape p esen s wo modelling amewo ks o as ime-domain simula ions o a ship
manoeu ing in wa es. The amewo ks a e de eloped based on he “ wo- ime scale” and “uni ied”
me hods. The wo- ime scale me hod sol es he low- equency and wa e- equency ship mo ions sepa-
a ely. The mean and slowly a ying d i o ces combined wi h he manoeu ing and appendages ones
a e applied o compu e he slowly a ying pa h. The wa e- equency mo ions a e hen sol ed h ough
he linea supe posi ion o esponse ampli ude ope a o s (RAOs) o each mo ion along his pa h. The
RAOs a e compu ed using he bounda y elemen po en ial low sol e SEACAL using Rankine sou ces.
The uni ied me hod on he o he hand sol es he low- and wa e- equency ship mo ions in one in e-
g a ed sys em using o ce supe posi ion. Imposing i s -o de wa e- equency o ces based on RAOs
in o a ime-domain simula ion can gene a e undesi ed (o ghos ) d i o ces. To o e come his, analy ic
ghos d i o ce a e p e-compu ed in SEACAL o la e co ec ion du ing he ime-domain simula ions.
Mo eo e , adia ion o ces in ime domain need o be p ope ly modelled in o de o a oid addi ional
spu ious d i o ces due o low equency damping componen s a ising om Rankine sou ce solu ions.
Fu he mo e, he pape compa es heading-keeping simula ions, using bo h me hods, wi h he expe i-
men s, showcasing he me hods’ capabili ies and sho comings. Fo his compa ison, he 5415M na al
comba an ship has been used as a case s udy o illus a e how he me hods pe o m in p ac ice. The
alida ion cases show ha he simula ions p edic he ship beha io in egula and i egula wa es wi h
ai ly good accu acy.
Key wo ds: Manoeu ing in wa es, wo- ime scale, uni ied, 5415M
1. In oduc ion
A eliable quan i ica ion o manoeu ing in wa es is o pa amoun impo ance o ship design and
pe o mance assessmen . This is mainly d i en by he Ene gy E iciency Design Index (EEDI) equi emen
o dec ease ins alled powe o me chan ships. The need has also inc eased due o equi emen s o heading
and ack-keeping, and o quan i y he con ollabili y in ad e se wea he , especially na al ships.
The pe o mance assessmen o ship manoeu ing in wa es can be s udied using nume ical simula ions
a ea lie design s age and physical model es s a a la e s age. The model has o be equipped wi h ac ua o s
(e.g. udde s and p opelle s) and au opilo o con ol he speed and heading. These es s a e challenging
no only o he model es s bu also o he nume ical simula ions. P ac ically, heading-keeping o ack-
keeping simula ions ha e o be pe o med wi h a du a ion o a leas 30 minu es up o 3 hou s wi h a
a ie y o wa e di ec ions and ship speeds. Those simula ions can only be pe o med e icien ly using a as
ime-domain simula ion ool.
In pe o ming ime-domain simula ions, wo amewo ks a e a ailable in li e a u e. The amewo ks a e
known as he uni ied and wo- ime scale me hods.
The uni ied me hods ha e been in oduced by W. R. McC eigh [1986], Hoo and Pie e s [1988], P.
A. Bailey, W.G. P ice, P. Tema el [1997]. The me hod sol es he equa ion o mo ions o he manoeu-
ing and seakeeping in one in eg a ed sys em. Con olu ion in eg al [Cummins, 1962] is used o e alua e
equency-domain based ans e unc ions o hyd odynamic coe icien s. The con olu ion in eg al equi es
an accu a e es ima ion o he impulse esponse unc ions. Fi s -o de exci a ion o ces a e compu ed in
*Co espondence o: [email p o ec ed]
1
ime-domain h ough he linea supe posi ion o he o ce ans e unc ions and wa e spec um ampli udes.
As also men ioned in [R. Skejic, 2013], he e alua ion o i s -o de o ces in ime-domain simula ions can
g adually gene a es nonlinea i ies ha can a ec he ship manoeu e s.
The wo- ime scale me hod was in oduced in [Skejic and Fal insen, 2008, H. Yasukawa and Y. Nakayama,
2009]. The me hod sol es equa ions o mo ion sepa a ely o di e en ime scale. I elies on p e-compu ed
esponse ampli ude ope a o s (RAOs) o he wa e- equency mo ions o he ship. These a e usually ob-
ained using equency-domain calcula ions. Then, mo ions in ime-domain a e compu ed in wo ollowing
s ages. Fi s ly, he low- equency su ge, sway, yaw and oll a e compu ed. This is done by conside ing
he manoeu ing o ces, including con ol su aces such as p opelle s and udde s, and mean second o de
wa e d i o ces. Then, he wa e- equency mo ions, o he six deg ees o eedom, a e compu ed a each
loca ion based on he p oduc o he RAO and he complex wa e ampli ude.
This pape p esen s he MARIN blending o manoeu ing and seakeeping models based on he uni ied
and wo- ime scale me hods. Analy ical o mula ions a e de i ed o emo e he un-physical non-linea i ies
(due o ghos d i o ces) ha a e gene a ed h ough he e alua ion o i s -o de exci a ion o ces and a-
dia ion o ces ans e unc ions in a ime-domain simula ion. Addi ionally, he pu e second-o de wa e
exci a ion o ces a e imposed bo h o he uni ied and wo- ime scale me hods h ough he ull quad a ic
ans e unc ions p o ided by a equency-domain bounda y elemen sol e wi h Rankine sou ce o mula-
ion. Bo h models use he ollowing ex a modelling aspec s: he Uni ied Damping model as de i ed in [F.
H. H. A. Quad lieg, R. Tonelli, and A. Bedos, 2021] o he manoeu ing coe icien s, he ou quad an s
p opelle app oach as in [Kuipe , 1992, J. Moulijn and R. Tonelli and U. Shipu ka , 2024] o he p opelle s’
h us and o que modelling, and MARIN’s udde model [Tonelli and Vogels, 2024].
The hyd odynamic coe icien s and wa e exci a ion o ces a e ob ained by sol ing double-body s eady
po en ial, adia ion and di ac ion po en ial p oblems. The eloci y po en ials a e ob ained om he sou ce
dis ibu ion solu ions o a bounda y elemen sol e wi h Rankine sou ce o e hull and ee-su ace domain.
The d i o ces quad a ic ans e unc ions (QTFs) a e compu ed wi h he quad a ic pa based on he
di ec p essu e in eg a ion me hod and he po en ial pa based on he Pinks e app oxima ion [Pinks e ,
1975]. The sol e is called SEACAL, de eloped unde Coope a i e Resea ch Ship amewo k (CRS 1)
[MARIN, 2024a,b].
Valida ion o he models a e pe o med agains he model es s o he 5415M na al comba an ship. The
model es s we e pe o med unde CRS p ojec . The da a ha e also been used in [F. H. H. A. Quad lied, S.
Rapuc, 2019] o alida e hei heading-keeping simula ions.
This pape is o ganised as ollows. The modelling amewo k o manoeu ing in wa es is desc ibed in
Sec ion 2.. Desc ip ion o he alida ion ma e ial is p esen ed in Sec ion 3.. Ve i ica ions o second-o de
exci a ion o ces and ghos d i o ce a e discussed in Sec ion 4., and he alida ions o heading-keeping
simula ions a e discussed in Sec ion 5.. Conclusions w ap-up his pape in Sec ion 6..
2. Modelling o manoeu ing in wa es
This sec ion desc ibes he modelling amewo ks o simula ing ship manoeu es in wa es. This sec ion
s a s wi h he desc ip ion o coo dina e sys ems and ans o ma ion. De i a ion o he equa ions o mo ion
and he modelling o hyd odynamic o ces, wa e exci a ion o ces, and he ac ua o s o ces a e desc ibed.
2.1. Coo dina e sys ems and ans o ma ion
Ship mo ions in six deg ees o eedom (DOF) a e desc ibed in h ee o hogonal coo dina e sys ems:
a global ine ial, body- ixed and seekeeping (equilib ium), ollowing he no a ion o Fossen [2005]. The
global ine ial ( ixed) coo dina e sys em is deno ed by {n}wi h (xn, yn, zn)axes and o igin on;xn,yna e
ho izon al axes pe pendicula o upwa d e ical axis zn. The body- ixed e e ence ame {b}is a mo ing
coo dina e ame wi h axis (xb, yb, zb)axis and o igin ob. The seakeeping ame {s}= (xs, ys, zs)is ixed
o he equilib ium s a e. The dis ance ec o s o he e e ence ames a e deno ed as s/n, b/s, and b/n
1Coope a i e Resea ch Ships (CRS) was s a ed in 1969 and ocuses on hyd odynamics, s uc u al and ela ed p oblems o all
kind o ship ypes om a undamen al, design and ope a ional pe spec i e. Today, he CRS consis s o 24 membe o ganiza ions and
companies ca ying ou a join wo k p og am. Mo e in o ma ion on h p://www.c ships.o g
2
o he dis ance ec o o {s}wi h espec o {n}, o {b}wi h espec o {s}, and o {b}wi h espec o
{n}, espec i ely. The ec o s sa is y he ollowing ela ion b/n = s/n + b/s.
The posi ion ec o o poin obwi h espec o {n}exp essed in {n}is deno ed as pn
b/n = (pE, pN, pU)
wi h he x-axis poin ing owa d eas , y-axis owa ds no h and z-axis upwa ds. A Eule angles ec o is
de ined as Θnb = (ϕ, θ, ψ). The ship mo ion in 6 DOF wi h he coo dina e o igin obcan be exp essed
in he gene alized ship posi ion and o ien a ion ec o ζ= (pn
b/n,Θnb)Tin which Tis he anspose
ope a o . Co espondingly, he body- ixed eloci y ec o is deno ed = ( b
b/n,ωb
b/n)T= ( 1, 2)T=
(u, , w, p, q, )Tand he o ce ec o is F= ( b
b/n,mb
b/n)T= (X, Y, Z, K, M, N)T.
The seakeeping e e ence ame {s}is conside ed ine ial ixed o an equilib ium s a e. The equilib-
ium s a e is de ined as n
s/n = (Ucos( ¯
ψ), U sin( ¯
ψ))T,ωn
s/n = (0,0,0)T,Θns = (0,0,¯
ψ)whe e U
is he ship speed and ¯
ψis he mean heading. The pe u bed mo ion is de ined as ξ= ( s
b/s,Θsb)T=
(ξ1, ξ2, ξ3, ξ4, ξ5, ξ6)T ha co esponds o su ge, sway, hea e, oll, pi ch and yaw pe u ba ions. The co e-
sponding eloci y and accele a ion in {s}a e hen exp essed as ˙
ξ,¨
ξ. The pe u bed eloci y exp essed in
{b}is deno ed as δ = ( b
b/s,ωb
b/s) = (δu, δ , δw, δp, δq, δ ).
Following [Fossen, 2005], a ans o ma ion be ween body and seakeeping ames can be ob ained as ol-
lows. The eloci ies in {s}and in {b} ela es h ough he ans o ma ion ma ix JΘ(ξ), o ming kinema ic
equa ion, as ˙
ξ=JΘ(ξ)δ (1)
whe e he ans o ma ion ma ix JΘ(ξ)is exp essed wi h he Eule angles o a ion ma ix Rs
b(Θsb) =
Rz,ξ6Ry,ξ5Rx,ξ4and he angula eloci y ans o ma ion ma ix TΘ(Θsb) o small angles as
JΘ(ξ) = Rs
b(Θsb)03×3
03×3TΘ(Θsb)whe e TΘ(Θsb) =
10ξ5
01−ξ4
0ξ41
.
The equilib ium linea eloci y in {b}is de ined as ¯
1=Rb
n(Θbn) n
s/n =Rb
s(Θbs)Ue1while he
angula eloci y ¯
2= (0,0,0)Tsince {s}do no o a e wi h espec o {n},ωs/n =
0. In small angles
app oxima ion, ¯
≈U(1,−ξ6, ξ5,0,0,0)T=U(e1−Lξ)and ˙
¯ ≈U(0,−δ , δq, 0,0,0)T=−ULδ
wi h L= [06×4,−e3,e2], wi h ejis he six dimension uni ec o . The ela ion o he igid body eloci y,
accele a ion and he Eule angles be ween he e e ences o ames a e hen exp essed as ollows
=¯
+δ =U(e1−Lξ) + δ ,˙
=˙
¯ +δ˙
=−ULδ +δ˙
,Θnb =Θns +Θsb.(2)
2.2. Equa ion o mo ions
Ship sailing a o wa d speed (U) encoun e s wa es wi h equency ωe=ω−kU cos(µ), whe e ω,k
and µa e espec i ely wa e equency, wa e numbe and wa e di ec ion. In equency domain, he ship
mo ions ec o is deno ed by ˆ
ξas a complex ampli ude o ξ. The equa ion o mo ion is hen exp essed in
e ms o igid body mass ma ices MRB, hyd odynamic coe icien s o added mass A(ωe)and damping
B(ωe), es o ing coe icien C, linea exci a ion o ces ˆ
F(1)
exc(ωe)and o he ex e nal o ces ˆ
Fex (ωe), as
ollows.
−ω2
e[MRB +A(ωe)] −iωeB(ωe) + Cˆ
ξ(ωe) = ˆ
F(1)
exc +ˆ
Fex .(3)
The hyd odynamic coe icien s and wa e exci a ion o ces a e ob ained om he equency-domain bound-
a y sol e , SEACAL.
In ime domain, he seakeeping equa ion ollows Cummins [1962] as
[MRB +A(∞)] ¨
ξ+Z
0
K( −τ)˙
ξ(τ)dτ +Cξ =Fwa e +Fex ,(4)
whe e he impulse esponse ( e a da ion) unc ion is de ined as
K( ) = 2
πZ∞
0
B(ωe) cos(ωe )dωe.(5)
3
Equa ion (4) ( ime-domain) ela es o Equa ion (3) ( equency-domain), h ough he Ogil e ans o ma ions
[T. F. Ogil e, 1964]. I can be shown ha he equency dependen added mass and damping coe icien s
can be ob ained om he e a da ion unc ion.
Equa ion (4) can be exp essed in body coo dina es {b}as
hMb
RB +Ab(∞)i(˙
+ULδ ) + Z
0
Kb( −τ)δ (τ)dτ +Cbξ=Fb
wa e +Fb
ex .(6)
The in luence o ξon he o wa d speed is o en neglec ed, so ha δ ≈ −Ue1. Equa ion (6) can be
ew i en wi h he mass ma ix Mb=Mb
RB +Ab(∞), Co iolis and cen ipe al o ces coe icien s due o
o a ion o {b}abou {s}C∗
b/s =MbUL, he hyd os a ic o ce Cbξ=Gζ and he s eady con ol o ce
Fb
sc =C∗
b/sUe1 o ob ain u=Uas
Mb˙
+C∗
b/s +Z
0
Kb( −τ) ( (τ)−Ue1)dτ +Gζ =Fb
wa e +Fb
ex +C∗
b/sUe1−Fb
sc.(7)
Equa ion (7) is a linea equa ion wi h he linea ized Co iolis and cen ipe al ma ices. Nonlinea i y can
be in oduced in Equa ion (7) by eplacing he Co iolis and cen ipe al ma ices o he one in {b}wi h
espec o {n}and adding non-linea damping D( ) o manoeu ing coe icien s. Fu he mo e, he wa e
d i o ces should also be aken in o accoun , so ha Fb
wa e =F(1)
exc +F(2)
exc. A uni ied seakeeping and
manoeu e ing model is hen exp essed wi h he kinema ic and dynamic equa ions as
˙
ζ=JΘ(ζ) ,(8)
Mb˙
+Cb/n( ) +D( ) +Z
0
Kb( −τ) ( (τ)−Ue1)dτ +Gζ =Fb
wa e +Fb
ex .
On he o he hand, wo- ime scale model sol es low- equency and wa e- equency mo ions sepa a ely.
The equa ion o mo ion o he low- equency mo ion is exp essed as
˙
ζLF =JΘLF (ζLF ) LF ,(9)
Mb˙
LF +Cb/n( LF ) LF +D(1) +D( LF ) LF +GζLF =F(2)
exc +Fb
ex .
in which he added mass con ibu ion in he mass ma ix Mband he Co iolis and cen ipe al ma ices
Cb/n, and he linea damping coe icien D(1) a e compu ed wi h ze o-encoun e equency. The pe u bed
posi ion and o ien a ion ec o δζ=ξis hen ob ained h ough a linea supe posi ion o he mo ion RAOs
(ob ained in Equa ion (3)) and wa e spec um componen s ˆηj, wi h he wa e equency ha co espond
wi h wa e numbe
k= (kx, ky)a ship posi ion p = (pE, pN), as ollows
ξ( ) = Re
Nω
X
j=1
ˆ
ξ(U, ωj, µ)ˆηjei(
kj·p−ωe )
.(10)
A ans o ma ion is needed o exp ess ξ= ( s
b/s,Θsb)in {n}, so ha ξn= (Rz, ¯
ψ s
b/s,Θsb). The inal
ship posi ion and o ien a ion ec o ζis hen gi en as,
ζ=ζLF +ξn.(11)
2.3. Blended model
Two a ian s o uni ied model can be de i ed om Eq. 8, u he called Blended and Fully blended
models. On he one hand, he Blended model compu es a linea hyd os a ic o ce based on he we ed hull
su ace a es in calm wa e . On he o he hand, he Fully blended model compu es non-linea hyd os a ic
o ce based on he ins an aneous we ed hull su ace due o he unpe u bed wa e ele a ion and ship mo ions.
This pape ocuses only o he Blended model, he Fully blended will be published in a u u e publica ion.
The ollowing subsec ions desc ibe he co ec ions o be applied o he Blended model.
4
2.3.1. Hyd odynamic coe icien s co ec ion
The accu acy o he e a da ion unc ions depends on he limi ing beha io o he hyd odynamic co-
e icien s (added mass and damping) a ze o and in ini e equencies. The hyd odynamic coe icien s a e
ob ained om SEACAL ha conside s he o wa d speed e ec in he double-body s eady po en ials. In
ha case, a p e-p ocessing be o e cons uc ing he e a da ion unc ions needs o be applied o he hyd ody-
namic da abase. A smoo hing unc ion is applied o he damping coe icien s so ha he alues go o ze o
o ω→0. This a oids an in e e ence wi h manoeu ing damping. Tha co ec ion is no necessa y when
he hyd odynamic da abase a e ob ained om he bounda y elemen sol e wi h ze o-speed G een- unc ion
me hod. Fu he mo e, maximum equency and equency s ep in he hyd odynamic da abase de e mine
also he accu acy o he e a da ion unc ion. Time s ep in he e a da ion unc ion could be oo coa se i
he maximum equency was oo small. While, he maximum ime o he e a da ion unc ion could be oo
sho i he equency s ep was oo coa se.
2.3.2. Ghos d i o ce emo al
In he blended models, he e alua ion o he i s -o de o ces in non-equilib ium s a e gene a es unde-
si ed non-linea o ces ha could signi ican ly in luence he manoeu ing pa h. The undesi ed o ces, ha
a e la e called ghos d i o ces, a e p oduced in he ollowing p ocess:
• The i s -o de exci a ion o ce is e alua ed wi h he phase o wa e a he ac ual posi ion.
• The i s -o de exci a ion o ce is e alua ed a he ac ual wa e- equency a ying heading.
• The hyd odynamic eac ion o ces due o added mass, damping and hyd os a ic a e compu ed wi h
ship- ixed mo ions, eloci y and accele a ions in he yawed sys em o axis including wa e equency
yaw mo ions.
Those h ee con ibu ions o ghos d i o ces a e la e e e ed as i s , second and hi d con ibu ions,
espec i ely.
As desc ibed in [T. Bunnik, 2022], he ghos d i o ce can be compu ed analy ically om equency-
domain bounda y elemen solu ions. The analy ical ghos d i o ces QTFs o he i s con ibu ion is
de i ed as ollows. Su ge and sway mo ions esponse o a bich oma ic wa e (ω1and ω2) a e exp essed wi h
he co esponding mo ion complex ans e unc ions ( ˆ
ξjj= 1 o su ge and j= 2 o sway) as
ξj=Re hˆ
ξj1e−iω1 +ˆ
ξj2e−iω2 i.(12)
The su ge and sway mo ions a e in {s}and hey accoun o he phase change o he wa e in {n}. The
i s -o de exci a ion o ce a he ac ual wa e phase (F(1∗)
exc ) is hen ob ained as, wi h he wa e numbe ec o
kl= (klcos µ, klsin µ),
F(1∗)
exc =Re hˆ
F(1)
exc1e−iω1 +i
k1·(ξ1,ξ2)+ˆ
F(1)
exc2e−iω2 +i
k2·(ξ1,ξ2)i.(13)
Linea izing he Equa ion (13) wi h an assump ion o small mo ion gi es
F(1∗)
exc ≈Re hˆ
F(1)
exc1e−iω1 1 + i
k1·(ξ1, ξ2)+ˆ
F(1)
exc2e−iω2 1 + i
k2·(ξ1, ξ2)i.(14)
The i s e m wi h he mul iplie one is he i s -o de exci a ion o ce e alua ed a he equilib ium s a e o
he body. The emainde is he second-o de o ce con ibu ion. The QTF o he i s con ibu ion ghos
d i o ce is hen ob ained by subs i u ing he su ge and sway mo ions Equa ion (12) in o Equa ion (14) and
conside ing only he mean and di e ence- equency con ibu ions.
The second con ibu ion o ghos d i o ce is de i ed as ollows. The i s -o de o ce a he ac ual
heading (ψ)wi h he assump ion o a small heading change a ound he mean heading can be app oxima ed
as
F(1∗)
exc (ψ) = F(1)
exc(¯
ψ) + δψ dF(1)
exc(¯
ψ)
dψ (¯
ψ) + O(δψ2).(15)
5
In a complex o m o a bich oma ic wa e, he o ce is hen ew i en, wi h
δψ =Re ξ61e−iω1 +ξ62e−iω2 ,(16)
as
F(1∗)
exc (ψ) = Re " ˆ
F(1)
exc1+δψ dˆ
F(1)
exc1
dψ !e−iω1 + ˆ
F(1)
exc2+δψ dˆ
F(1)
exc2
dψ !e−iω2 #.(17)
The i s e ms, ˆ
F(1)
exc1and ˆ
F(1)
exc2, a e he co ec i s -o de o ces. The QTFs o he ghos d i o ces a e
hen ob ained by subs i u ing Equa ion (16) in o he emainde e ms o Equa ion (17) and conside ing only
he mean and di e ence- equency con ibu ions.
The hi d con ibu ion o d i o ces a e de i ed as ollows. In ou ime-domain simula ion ame-
wo k, he adia ion o ces and es o ing o ces a e e alua ed wi h he body ixed mo ions, eloci ies,
and accele a ion in he yawed e e ence ame. The eloci y and accele a ion ec o s a e deno ed by
yawed = ( ˙x(¯
ψ),˙y(¯
ψ),˙z, ˙
ϕ(¯
ψ),˙
θ(¯
ψ),˙
ψ)and ˙
yawed = (¨x(¯
ψ),¨y(¯
ψ),¨z, ¨
ϕ(¯
ψ),¨
θ(¯
ψ),¨
ψ), espec i ely. The
eloci y and accele a ion can be ans o m o {b}as ollows
b=R−1
z,ψ yawed,˙
b=R−1
z,ψ ˙
yawed,whe e R−1
z,ψ =Rz,ψ 03×3
03×3Rz,ψ −1
.
The eac ion o ces a e hen exp essed as
Fb
ad =AR−1
z,ψ ˙
yawed +BR−1
z,ψ yawed (18)
By aking linea iza ion wi h a small yaw angle assump ion, he adia ion o ce Equa ion (18) con ains i s -
and second-o de con ibu ion. The QTFs o he ghos d i o ce can hen be ob ained by applying he
second-o de o ce con ibu ion o Equa ion (18) in a bich oma ic wa e wi h a complex o m o mula ion.
Following simila app oach as abo e, he second-o de con ibu ion o he es o ing o ce and he co e-
sponding QTFs can be de i ed.
The o al ghos d i o ces QTFs a e hen used in he ime-domain simula ion o he o ce co ec ion.
The wa e exci a ion o ce o he blended model becomes
FbB
wa e =F(1)
exc +F(2)
exc −F(2)
ghos .(19)
2.4. Manoeu ing eac ion o ces and calm wa e esis ance
The manoeu ing eac ion o ces a e modelled ia coe icien s o i p e-de e mined o ces and mo-
men s, ia he so-called Uni ied Damping Model (UDM). The UDM models he manoeu ing eac ion
o ces (damping and added ine ia o ces due o a mo ion) ia coe icien s mul iplied by he mo ion de i a-
i es (ship eloci ies and accele a ions). These coe icien s i cap i e manoeu ing model expe imen o
CFD esul s, ollowing he p ocedu e desc ibed in [V. Fe a i, R. Tonelli, A.S. Kisjes and R. Hallman, 2022].
The Uni ied Damping Model conside s a s a boa d-po side symme y o he ship o loa ing s uc u e.
Calm wa e (sailing s aigh ) esis ance da a om model es s is imposed o he simula ions. In e pola-
ion o he da a o e he ship speed is applied in he simula ions.
2.5. P opelle and udde o ces
2.5.1. P opelle
The p opelle h us and o que modelling equi es inpu o a p opelle open wa e cu e, h us deduc-
ion ac o and wake ac ion. Fi s , he ins an aneous ad ance eloci y (VA) a he p opelle which includes
he wake ac ion (w) and he ship eloci y (V) is compu ed as VA=V(1 −w). The ad ance a io (J) is
hen compu ed o a gi en p opelle RPM (n) and a diame e o p opelle (D) as J=VA
nD .
In he egula open wa e cha ac e is ic me hod, he h us (T) and o que (Q) a e compu ed wi h he
h us (KT) and o que (KQ) coe icien s as a unc ion o he ad ance a io J. Howe e , his app oach is
limi ed o posi i e o a ion a e and posi i e ad ance speed. Di e en ly, a ou quad an s ep esen a ion
6
o he p opelle open wa e cha ac e is ic can model p opelle h us and o que o any combina ion o
p opelle o a ion a e and ad ance speed. This me hod has been implemen ed and desc ibed in [J. Moulijn
and R. Tonelli and U. Shipu ka , 2024, Kuipe , 1992].
In he ou quad an s app oach, he h us and o que a e compu ed wi h he h us (CT) and o que (CQ)
coe icien s as unc ion o he p opelle pi ch angle (β∗). The p opelle pi ch angle is hen de ined as an
angle be ween he ad ance and o a ional eloci ies. The o a ional eloci y (VT) o he blade is compu ed
o ins ance a sec ion 0.7.R, whe e he p opelle adius R=D/2, as VT= 0.7πnD. The pi ch angle is
hen exp essed as β∗= a c an VA
VT. The pi ch angle anges om -180 deg o 180 deg and is di ided
in o ou quad an s. The h us , ans e sal o ce, and o que a e hen compu ed wi h he in low eloci y
VR=pV2
A+V2
Tas ollows
T=1
8CT(β∗)ρV 2
RπD2, Y =1
8CY(β∗)ρV 2
RπD2, Q =1
8CQ(β∗)ρV 2
RπD3.(20)
2.5.2. Rudde
In o de o main ain he ship on he ack o on heading, he udde is s ee ed by an au opilo . The
udde ma hema ical model desc ibes he low a ound he udde , a e aging eloci y and o ce componen s
in a single poin o applica ion. The model is de ined by i s dimensions, loca ion, and s ock inclina ion angle
in he y-z plane. The model includes he e ec o an accele a ed in low when posi ioned in a p opelle s slip
s eam, and he e ec o a s aigh ened in low due o he p esence o he hull. The ma hema ical model
is based on li ing line and axial momen um (ac ua o disc) heo y, wi h he implemen a ion o empi ical
coe icien s based on wind unnel and owing ank model es s. The model heo y and alida ion is ully
desc ibed in [Tonelli and Vogels, 2024].
2.5.3. Con olle
Heading-keeping con olle is applied in simula ions, p esen ed in Sec ion 5., wi h he au opilo gains o
2 [-], 0.018 [1/s] and 0.549 [s] o he p opo ional, in eg a o and damping gains, espec i ely. Addi ionally
a speed con olle is applied wi h 0.2 [-], 0.01 [1/s] and 0 [s] o he p opo ional, in eg a o and damping
gains, espec i ely. This speed con olle is also es ic ed by a p opelle con olle when he maximum
powe is exceeded. The maximum powe is p ede ined om an accele a ion es o each a a ge speed.
3. Valida ion ma e ial
This sec ion desc ibes he model es da a used o he alida ion o he nume ical models.
In he CRS MORE wo king g oup, manoeu ing and seakeeping expe imen s wi h a model o he
5415M we e pe o med a MARIN. The 5415M is a published na al comba an ha is widely used in
manoeu ing s udies. The CRS MORE model es campaign includes he ollowing es s: i) Cap i e ma-
noeu ing expe imen s, epo ed in [R. Tonelli, 2020a] ii) F ee sailing manoeu ing expe imen s, epo ed
in [R. Tonelli, 2020b], iii) Seakeeping expe imen s, epo ed in [S. Rapuc, 2020]. Addi ionally, he cap i e
es s da a ob ained in [R. Hallman, 2007] we e used o make he UDM.
An o e iew o he model is shown in Figu e 1.. The model is appended wi h p opelle s, udde s, bilge
keels and in s abilize s. The ship pa icula s a e p esen ed in Table 1.. The seakeeping model es s we e
pe o med wi h a heading- and ack-keeping au opilo . In his pape , only he da a o heading-keeping
es s wi h ac i e in s abilize s will be used o he alida ion es . An o e iew o he seakeeping es s is
p esen ed in Table 2. o egula wa es cases and Table 3. o i egula wa es wi h Pie son-Moskowi z
spec a.
7
Figu e 1. O e iew o 5415M model
Table 1. Main pa icula s o he 5415M hull
DESIGNATION SYMBOL MAGNITUDE UNIT
Leng h be ween pe pendicula s LP P 142 m
B ead h moulded on wa e line BW L 19.06 m
D augh moulded on FPTF6.15 m
D augh moulded on APTA6.15 m
Displacemen olume moulded ∇8428.6 m3
T ans e se me acen e heigh abo e baseline KM 9.45 m
Ve ical cen e o g a i y ( om keel line) VCG 7.51 m
T ans e se me acen ic heigh GM 1.94 m
Longi udinal cen e o buoyancy ( o e o AP) LCB 70.05 m
App oxima e Roll adius o gy a ion kxx 7.62 m
App oxima e Pi ch adius o gy a ion kyy 35.5 m
App oxima e Yaw adius o gy a ion kzz 35.5 m
Table 2. Regula wa e es s wi h heading-keeping au opilo and ac i e ins
Wa e Ampli ude [m] wa e equencies [ ad/s] Wa e di ec ions [deg] Ship speed [kn]
1 0.45, 0.55, 0.65, 0.9,1.2
1.75 0.45, 0.55, 0.65, 0.9,1.2 180, 135, 60, 30 4, 12, 20
3.0 0.55, 0.9
Table 3. I egula wa e es s wi h heading-keeping au opilo and ac i e ins
Sea s a e Hs [m] Tp [s] γWa e di ec ions [deg] Ship speed [kn]
SS-3 1.25 7.5 180, 135, 60, 30, 0 4
SS-4 2.5 8.8 1 180, 135, 60, 30 12, 20
SS-6 6.0 12.4 180, 135, 60, 30 4, 12, 20
8
4. Ve i ica ion o ghos d i o ce
A so -moo ing sys em was se up using linea sp ing coe icien s applied o su ge, sway and yaw
momen . The es o ing coe icien s a e se up so ha he co esponding na u al pe iods a e 140 s, 120 s
and 100 s o su ge, sway and yaw mo ions, espec i ely. A linea sp ing damping o 7.5% o he c i ical
damping in each mode was used o he s abili y. The simula ions we e pe o med in egula wa es o e
a ange o wa e equency in (0.2,1.6) ad/s wi h ampli ude 1 m. Fo each simula ion, a s able o se is
measu ed and hen d i o ces a e compu ed by mul iplying he es o ing coe icien wi h he co esponding
o se o each mode.
Time-domain simula ion was se up wi h he ollowing o ces ac ing on he ship: linea hyd os a ic
o ce, adia ion o ce, i s -o de exci a ion o ce and he moo ing o ce. The compu ed QTFs ob ained
om he ime-domain simula ions (XMF) a e compa ed agains he da abase ob ained using SEACAL. The
esul s a e shown in Figu e 2. o ze o-speed case in head and beam seas and in Figu e 3. o he ship sailing
a 4 kno s in beam seas.
The simula ions showed ha he second-o de d i o ces a e gene a ed by only imposing i s -o de
exci a ion, linea hyd os a ics and adia ion o ces. This con i ms he exis ence o ghos d i o ces in he
ime-domain simula ions ha ag ee wi h he analy ic o ces compu ed in SEACAL.
Compa ing he ghos and pu e d i o ces, while he ghos d i o ces a e much smalle wi h espec
o he d i o ces, he yaw ghos d i momen can be equal o la ge han he pu e d i momen . This
sugges ha he ghos d i momen canno be igno ed. Disc epancies o yaw d i momen be ween XMF
and SEACAL a highe equencies a e obse ed. This is p obably due o he coupled mo ions con ibu ions
which we e no conside ed in he d i o ce calcula ion.
Figu e 2. Compa ison o he d i o ces be ween SEACAL and XMF a ze o-speed in head seas (180 deg) and beam
seas (90 deg). The op plo shows he su ge d i o ces in head seas. The middle and bo om plo s show he sway d i
o ces and yaw d i momen s in beam seas.
9
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